On the long-time asymptotics of quantum dynamical semigroups (Q2888957)

Let \(\mathcal{M}\) be a von Neumann algebra. A family of normal, unital, positive linear maps \(\alpha_t:\mathcal{M}\to\mathcal{M}\) \((t\geq0)\) such that \({\alpha_t}\circ\alpha_s=\alpha_{t+s}\) is called a quantum dynamical semigroup. If each \(\alpha_t\) is completely positive, then the semigroup is called CP. A projection \(p\) is called sub-harmonic with respect to a \(\alpha_t\) if \(\alpha_t(p)\geq p\) and it is called sub-harmonic if \(\alpha_t(p)\leq p\).NEWLINENEWLINEIn this note, the authors show that the set of sub-harmonic and the set of super-harmonic projections with respect to \(\alpha_t\) \((t\geq0)\) are both complete lattices. Also, for a finite-dimensional von Neumann algebra \(\mathcal{M}\), they show that, if \(\{\alpha_t:t\geq0\}\) is CP, then \(\sup\{\alpha_{t}(r_0):t\geq0\}=1\), where \(r_0\) is the smallest projection.NEWLINENEWLINEFor the entire collection see [Zbl 1238.81004].